Efficient solution of convection-dominated and diffusion-reaction problems via a novel eighth-order iterative method

[EN] High-order iterative schemes are theoretically faster than the classical Newton method, but their practical performance is often limited by high computational cost or by narrow stability regions near singularities. This paper presents ARJ8S, a new eighth-order vectorial algorithm that achieves high convergence order with low computational cost and good stability. Crucially, we prove that the ARJ8S's multi-step architecture can be easily generalized using multi-stage backward linear searches ruled by Armijo-type conditions. This globalization mechanism protects the iterative scheme from local overfitting, preventing catastrophic exponential blow-ups near bifurcation points, where standard high-order schemes often produce NaN values. The scalability and validity of the proposed framework are tested on convection-dominated problems, such as the viscous Burgers' equation, and on massive 2D diffusion-reaction Bratu systems. To the best of our knowledge, this work represents one of the scarce instances in the literature where an iterative scheme exceeding the fourth order of convergence successfully solves a more than four-million unknowns problem (d > 4.1 & times; 10(6)) under heavily ill-conditioned singular boundaries. In these extreme dimensions, and in regimes where sparse direct factorizations remain feasible, the stabilized ARJ8S-Armijo framework halves the required global iteration steps, delivering up to a 30.4% computational speedup over the classical Newton method while using frozen sparse factorizations and a strictly shared weight function.